A fundamental measure theory for the sticky hard sphere fluid

0.33 MB·English

by  Hendrik Hansen-Goos

A fundamental measure theory for the sticky hard sphere fluid
Hendrik Hansen-Goos1,∗ and J. S. Wettlaufer1,2,3
1Department of Geology and Geophysics, Yale University, New Haven, CT 06520, USA
2Department of Physics, Yale University, New Haven, CT 06520, USA
3Program in Applied Mathematics, Yale University, New Haven, CT 06520, USA
(Dated: January 10, 2011)
We construct a density functional theory (DFT) for the sticky hard sphere (SHS) fluid which,
like Rosenfeld’s fundamental measure theory (FMT) for the hard sphere fluid [Phys. Rev. Lett.
63, 980 (1989)], is based on a set of weighted densities and an exact result from scaled particle
theory (SPT). It is demonstrated that the excess free energy density of the inhomogeneous SHS
1 fluidΦSHS isuniquelydefinedwhen(a)it issolely afunction oftheweighted densitiesfrom Kierlik
1 andRosinberg’sversionofFMT[Phys. Rev. A42,3382(1990)],(b)itsatisfiestheSPTdifferential
0
equation,and(c)ityieldsanygivendirectcorrelation function(DCF)from theclass ofgeneralized
2
Percus-Yevick closures introduced by Gazzillo and Giacometti [J. Chem. Phys. 120, 4742 (2004)].

n TheresultingDFTisshown tobein verygood agreement with simulation data. Inparticular, this
a FMTyieldsthecorrect contactvalueofthedensityprofileswithnoadjustableparameters. Rather
J than requiring higher order DCFs, such as perturbative DFTs, our SHS FMT produces them.

7 Interestingly, although equivalent to Kierlik and Rosinberg’s FMT in the case of hard spheres, the
set of weighted densities used for Rosenfeld’s original FMT is insufficient for constructing a DFT

] which yields theSHSDCF.
t
f
o
s
.
t
a
I. INTRODUCTION
m
-
d Density functional theory (DFT) was developed to characterize quantum systems in an external potential [1] and
n since the 1970’s has been fruitfully brought to bear on classical fluids [2]. Indeed, here too it serves as an excellent
o tool to study the microscopic structure of fluids under the influence of external potentials, such as confining walls,
c and can serve as the sole prerequisite to map out complete phase diagrams that encompass various fluid and crys-
[
talline structures. Moreover,besides yieldinga soundtheoreticalunderstandingoffindingsfromcolloidalphysicsand

2 computer simulations, DFT calculations can be extremely efficient computationally, especially if the systems under
v consideration allow for dimensional reduction, and hence in consequence largely outperform Monte Carlo (MC) sim-
2 ulations. In particular, DFT is especially convenient when phase diagrams or solvation free energies are calculated
7
because it makes direct use of the grand potential Ω. The basic formalism of DFT expresses Ω as a functional of the
7 density profiles ρ (r) of the different fluid components i=1,...,ν as
1 i
.
0 ν
1 Ω[ ρ (r) ]= [ ρ (r) ]+ [ ρ (r) ]+ drρ (r)[Vext(r) µ ], (1)
0 { i } Fid { i } Fex { i } Xi=1Z i i − i
1
: where µ is the chemicalpotentialofspecies i, andVext(r) the externalpotentialacting onparticlesof speciesi. The
v i i
i ideal gas part of the free energy is
X
ν
r
a β [ ρ (r) ]= drρ (r)(ln[Λ3ρ (r)] 1), (2)
Fid { i } Z i i i −
Xi=1
where Λ is the thermalwavelengthof species i and β =(k T)−1. The excesspartof the free energy [ ρ (r) ] de-
i B ex i
F { }
pendsontheinteractionsbetweentheparticlesandontemperature. Forthegivenexternalpotentials,theequilibrium
density profiles are obtained by solving the system of coupled equations δΩ/δρ 0, i=1,...,ν.
i
≡
The aboveformalismis exactsubjectto the limitationsofknowledgeregarding for whichwehaveonlyapprox-
ex
F
imate descriptions for 3D fluids with non-trivialinteractions. Among these approximations,Rosenfeld’s fundamental
measure theory (FMT) for the hard-sphere (HS) fluid [3] provides an excellent framework that uses only exact re-
sults for certain limiting cases which are generalized using dimensional analysis of a set of weighted densities. The
theory applies equally to mixtures and, in part thanks to subsequent incorporations of highly accurate equations of
∗Electronicaddress: [email protected]
2
state [4, 5], compares extremely well with computer simulations. The theory can be extended to treat crystallization
of the HS fluid [6–8] as well as the isotropic-nematic transition of non-spherical hard particles [9–12]. All of these
developments make FMT a central and powerful tool for the study of entropy-governedhard-body fluids.
Hard-body systems are an important test bed for theories which make use of a Weeks-Chandler-Andersen [13]
type decomposition of the interaction potential into a repulsive core and an attractive long-ranged tail. Moreover,
they can often be considered as an accurate approximation to an entropy-dominatedsystem. However, indeed many
realistic fluids that occur in biological applications or colloidal science invariably exhibit attractive particle-particle
interactions in addition to the hard-core repulsion. A simple model fluid with an attractive potential that has been
extensively studied by theorists [14, 15] is the square-well(SW) fluid defined by an interaction potential
, 0 r<σ
∞ ≤
βφ (r)= βǫ , σ r σ′ (3)
SW − SW ′ ≤ ≤
0, r >σ
′
where σ =2R is the particle hard-core diameter, σ σ is the range of the attraction and ǫ is the well depth.
SW
−
The radialdistributionfunction(RDF) g(r) andthe directcorrelationfunction(DCF) c(r) arerelatedthroughthe
Ornstein-Zernike (OZ) integral equation [16]
g(r) 1=c(r)+ρ dr′c(r′)(g(r r′ ) 1). (4)
− Z | − | −
However, the OZ equation can only be solved numerically within the Percus-Yevick (PY) closure for the potential
φ [17]. Alternatively, g(r) can be determined in computer simulations [18] from which c(r) and thermodynamic
SW
properties of the SW fluid can be derived. However, these results cover only a limited selection of specific parameter
choices for the model simulations and the results are clearly not in a convenient form for analysis. Hence, it is
reasonable to seek a model with attractive interactions that would be solvable within the usual closure relations for
the OZ equation. In the limiting case of surface adhesion where
σ
′
σ σ 0 and βǫ =ln , (5)
− → SW 12τ(σ′ σ) →∞
−
Baxter’s so-called sticky limit, with τ−1 characterizing the strength of the attraction, allows the OZ equation to be
solved within the PY closure [19]. The results are convenient in a closed analytical form and compare well with
computer simulations for various attraction strengths τ−1 [20]. Essential features of the sticky hard sphere (SHS)
fluid, such as the existence of a critical point associated with a liquid-gas transition, are captured by the PY results
[21]. Subsequently, the model has been generalized for SHS mixtures where the PY closure leads to a set of coupled
quadratic equations that can be solved analytically only in special cases [22, 23].
Far from being just a somewhatartificialtheoreticalconstruct, the SHS fluid has been demonstratedto provide an
accurate descriptionof colloidalsystems with very shortrangedattractions[24–26] and has been applied successfully
to phenomena such as structural arrest in a micellar system [27], competitive protein adsorption [28], and even skim
milkacidification[29]. Thatsaid,the useoftheSHSfluidtomodelexperimentalsystemshasbeenaimedatmapping
outphasediagramsandcalculatingequationsofstatewhiletheinhomogeneousfluidhasbeenstudiedlessthoroughly.
Firstly, this has to do with the difficulty of measuring structural properties such as density profiles and higher order
correlationfunctions. Secondly,welackareliabletooltocomputethesequantitiesbasedontheanalyticalPYresults.
ExistingDFTsforBaxter’sSHSfluidareallbasedonanexpansionoftheone-particleDCFinpowersofthedeviation
ofthedensityprofilefromthebulkdensity. Thisexpansioncanonlybeperformedsystematicallytolinearorderwhere
it involves the DCF for which the PY results can be used [30]. Moreover, the quality of the resulting DFT is rather
poor, especially close to contactwith a hardwall [30]. Expanding the one-particle DCF to quadraticorder requires a
formula for the unknown three-particle DCF c(3) of the SHS fluid. The functional form of c(3) has been estimated by
different authors based on assumptions regarding its range [31, 32] or its factorization in terms of c(r) [33]. In order
to yield reasonable agreement with computer simulations all these extensions of the DFT by Choudhury and Ghosh
[30] requireanadditional,density dependent, fitting parameterwhichis chosensuchthat the correctcontactvalue of
the density profile or the correct bulk pressure is guaranteed.
Here we develop a DFT for the SHS fluid that is distinct from the above perturbative approaches. In the spirit of
FMT we use a set of weighted densities that can be identified from a systematic inspection of the HS fluid in the low
density limit. In orderto constructa functionalwhichyields the PY DCF forthe SHS fluid, we supplementthis with
an exact relation from scaled particle theory (SPT) [34, 35]. The main result is a unique construction for a broad
class of closure schemes of which the PY closure is but one. No additional input and no fitting parameters are used.
Rather than requiring higher order DCFs our theory produces them.
3
InSec.IIwepresentthegeneralizedPYclosureasintroducedbyGazzilloandGiacometti[36]. Wethensummarize
in Sec. III the FMT for the HS fluid in the version developed by Kierlik and Rosinberg (KR) [37, 38]. In Sec. IV we
construct the new FMT for the single component SHS fluid, and then compare the results from the new functional
with data from numerical simulations in Sec. V. In the concluding section we place the new functional in a broader
context,discussingbothits meritsandlimitationswitha specialfocusonfuture developmentsthatmightbe pursued
in the framework of our approach.
II. GENERALIZED PERCUS-YEVICK APPROXIMATION
Gazzillo and Giacometti introduced the generalized Percus-Yevick (GPY) approximation for the SHS fluid [36].
The GPY is defined by a closure relation for the DCF c(r) of the form
[1+γ(r)], 0<r <σ,
−
c (r)=c (r), σ <r <σ′, (6)
GPY  shrink ′
0, r >σ .

Hereγ(r)=ρ dr′c(r′)h(r r′ )istheconvolutionoftheDCFwiththetotalcorrelationfunctionh(r)=g(r) 1,ρis
| − | −
the particle nuRmber density of the bulk fluid. Depending on the choice of the function cshrink(r), when applied to the
SHS fluid the GPY reduces to different familiar closure relations such as the mean spherical approximation (MSA),
the modified mean spherical approximation (mMSA), and the PY closure [36]. For example, choosing c (r) =
shrink
f(r)[1+γ(r)]inEq.(6),wheref(r)istheMayerf-function,resultsinthePYclosurerelationc (r)=f(r)[1+γ(r)]
PY
for arbitrary r >0.
Gazzillo and Giacometti [36] showed that in Baxter’s sticky limit the GPY approximation allows for a general
analytical solution of the OZ equation where the underlying closure relation enters solely through its approximation
to the cavity function y at contact. The resulting DCF is
σ
σ 1 r η r 3
c (r)= 12ηq2 a2+12η (a+b)2 aq a2 Θ(σ r)+q σδ(r σ), (7)
GPY (cid:26)− σr − (cid:20)2 − σ(cid:21)σ − 2 (cid:16)σ(cid:17) (cid:27) − σ −
where
1+2η 12q η 3η 6q η y
σ σ σ
a= , b= + , and q = . (8)
(1 η)2 − 1 η −2(1 η)2 1 η σ 12τ
− − − −
The PY closure corresponds to taking
2yHS(η)
yPY(η)= σ , (9)
σ
1+ η + 1+ η 2 ηyσHS(η)
(1−η)τ r(cid:16) (1−η)τ(cid:17) − 3τ2
in which
1+η/2
yHS(η)= (10)
σ (1 η)2
−
isthePYresultforthe HSfluid. ThePYresultfory [Eq.(9)]isexactuptolinearorderinη,withasmalldeviation
σ
occurring at quadratic and higher orders.
It turns out that the MSA leads to yMSA = 0 and hence does not capture the effect of surface adhesion in the
σ
SHS fluid and thus the results coincide with those for the HS fluid. The mMSA has ymMSA = 1 which is the exact
σ
result in the low density limit; η 0. Gazzillo and Giacometti [36] also introduce a C1 approximation wherein
yC1 = 1+ 5 1 + 1 η, and he→nce it corresponds to the exact expression for y truncated after the linear term
σ 2 − τ 12τ2 σ
in η. (cid:0) (cid:1)
III. FUNDAMENTAL MEASURE THEORY FOR THE HARD SPHERE FLUID
Presently, FMT provides the most accurate free energy model for inhomogeneous HS fluids. Additionally, FMT
gives anexcellent descriptionofHS mixtures [39] as well as fluids containing non-sphericalhardparticles [9–12]. The
4
theoryoriginatedwithRosenfeld’sseminalpaper[3]andhasbeencontinuouslymodifiedbyimprovingtheunderlying
equation of state [4, 5] and the ability to capture the formation of HS crystals [6–8]. There is an equivalent FMT
formulation due to Kierlik and Rosinberg [37, 38], which, in the case of the SHS fluid proves to be more useful than
Rosenfeld’s FMT (cf. Sec. VI). In order to make this development reasonably self-contained, we summarize KR’s
version of FMT here.
Fundamental measure theory provides an expression for the excess free energy which can be used in the
ex
F
framework of classical DFT to obtain the structural properties (density profiles and correlation functions) of fluids
that are subject to external potentials, i.e., inhomogeneous fluids. As discussed above, we consider a fluid with ν
different components which have the corresponding density profiles ρ (r), i = 1,...,ν. The FMT approach is based
i
on weighted densities n (r) written as
α
ν
n (r)= dr′ρ (r′)ω(α)(r r′) (11)
α Z i i −
Xi=1
in terms of the weight functions ω(α) of which there are four in KR’s approach corresponding to four n . These can
i α
be described in Fourier space:
ω(3)(k)=4π[sin(kR ) kR cos(kR )]/k3, (12)
i i − i i
ω(2)(k)=4πR sin(kR )/k, (13)
i i i
ω(1)(k)=[sin(kR )+kR cos(kR )]/2k, and (14)
i i i i
ω(0)(k)=cos(kR )+(kR /2)sin(kR ), (15)
i i i i
or in real space as
ω(3)(r)=Θ(R r), (16)
i i−
ω(2)(r)=δ(R r), (17)
i i−
ω(1)(r)=δ′(R r)/8π, and (18)
i i−
ω(0)(r)= δ′′(R r)/8π+δ′(R r)/2πR δ(R r)/2πR2. (19)
i − i− i− i− i− i
Here, Θ is the Heaviside function and the primes denote differentiation. Using these weighted densities, the FMT
excess free energy for the hard sphere mixture is readily obtained as
β [ ρ ]= drΦ ( n (r) ), (20)
ex i HS α
F { } Z { }
where the excess free energy density Φ is given by
HS
n n n3
Φ = n ln(1 n )+ 1 2 + 2 . (21)
HS − 0 − 3 1 n 24π(1 n )2
3 3
− −
Fundamental measure theory, which may be derived independently of PY integral theory, can be shown to yield the
same equation of state and the same DCF as obtained using the PY closure.
IV. FUNDAMENTAL MEASURE THEORY FOR THE STICKY HARD SPHERE FLUID
The FMT excess free energy density Φ obeys the SPT differential equation
HS
∂Φ ∂Φ
=n Φ+ n , (22)
0 α
∂n − ∂n
3 Xα α
which insures that the reversible work of adding a sphere to the fluid, in the limit of infinite sphere size, equals the
volume of the sphere multiplied by the pressure of the bulk fluid [34, 35]. While this exact relation must hold for
spheres with surface adhesion, there is no straightforwardgeneralizationof the principle for fluids with a finite range
ofinteraction(inadditionto the hardbody repulsion)suchas in the case ofa SW fluid. Forthese fluids the question
arises as to how the range of the interaction should scale as the size of the added sphere goes to infinity. However,
5
for the SHS fluid the surface attraction is subdominant with respect to the sphere volume in the large sphere limit.
Therefore, Eq. (22) must be obeyed.
Requiring Eq. (22) to hold for the SHS fluid of radius R and Baxter parameter τ implies that the corresponding
free energy density Φ must have the form
SHS
Rn R−2n Rn
Φ =Φ +R−1n φ 2 + 2 φ 2 , (23)
SHS HS 1 1 2
(cid:18)1 n (cid:19) 2π (cid:18)1 n (cid:19)
3 3
− −
where φ and φ are arbitrary dimensionless functions of the dimensionless argument. We have made the additional
1 2
assumption that Φ is linear in n which follows from inspection of the DCF c obtained from an excess free
SHS 1 FMT
energy density Φ that is solely a function of the weighted densities n . Then, in Fourier space we have
α
∂2Φ
c (k)= ω(α)(k)ω(β)(k), (24)
FMT
− ∂n ∂n
Xα,β α β
where,becausewewilltreatonlythesinglecomponentfluid,wehavedroppedthesecondindexoftheweightfunctions.
From the definition of the weight functions [Eqs. (12) – (15)] it can be seen that the inverse Fourier transform of
ω(1)(k)ω(1)(k) does not converge and hence, to insure that the DCF is defined in real space, we require Φ to be
SHS
linear in n . A similar argument leads us to require Φ Φ to be independent of n .
1 SHS HS 0
−
A crucial observation is that for any closure of the GPY class, there is a unique choice for the functions φ and
1
φ in Eq. (23) such that c [Eq. (24) using Eq. (23)] equals c from Eq. (7). This can be shown by comparing
2 FMT GPY
c and c in Fourier space.
FMT GPY
The functions φ and φ are obtained through integration of the cavity function at contact y associated with the
1 2 σ
given GPY closure. This is accomplished using the equations
′
φ (x)= 2y˜ (x)/x, (25)
1 − σ
y˜ (x)2
[x2φ′(x)]′ = σ xy˜ (x)+xy˜′(x), (26)
2 2 − σ σ
where x= Rn2 and hence in the bulk fluid, where n =η and n =3η/R, we have x= 3η or η = x . The latter
1−n3 3 2 1−η 3+x
equation is used to rewrite y˜ = ηy /τ as a function of x rather than as a function of η. The constants associated
σ σ
with the solution of Eqs. (25) and (26) are determined from the condition that φ and φ must vanish as x 0.
1 2
The PY closure is achieved by using yPY from Eq. (9) in the formulas above and we find →
σ
φPY(x)=12τ √X 1 4x 12τlnY +2√2(6τ 1)lnZ, and (27)
1 (cid:16) − (cid:17)− − −
36τ3 x2
φPY(x)= √X 1 6τ2 5√X 3 +τ 3+9x √X x√X +
2 x (cid:16) − (cid:17)− (cid:16) − (cid:17) (cid:16) − − (cid:17) 6
1
+36τ2lnY + 1 54τ2 lnZ, (28)
√2 −
(cid:0) (cid:1)
where
x 2 x x
X = 1+ 1+ , (29)
(cid:16) 3τ(cid:17) − 9τ2 (cid:16) 2(cid:17)
1 x x
Y = √X +1 + , and (30)
2(cid:16) (cid:17) 6τ − 36τ2
x−1 +2+√2X
Z = 3τ . (31)
2+√2 1
− 3τ
The MSA for the SHS fluid has yMSA =0, and hence φ =φ =0, giving identical results to those for the HS fluid
σ 1 2
without surface adhesion. The mMSA and the C1 approximation can be summarized by using the low density limit
expression yLDL = 1+y η with y = 0 to recover mMSA and y = 5 1 + 1 for the C1 approximation. Solving
σ 1 1 1 2 − τ 12τ2
6
Eqs. (25) and (26) with yLDL yields the functions
σ
2 y x x
φLDL(x)= 1 (1+y )ln 1+ , and (32)
1 τ (cid:20)3+x − 1 (cid:16) 3(cid:17)(cid:21)
1 x
φLDL(x)= 6+x+6τ(2+x)+2y (9+x+3τ(7+2x))+y2(12+x) ln 1+
2 2τ2x 1 1 3
(cid:2) (cid:3) (cid:16) (cid:17)
1 y
(2+4τ +τx) 1 18+42τ +5x+19τx+τx2
− 2τ2 − 2τ2(3+x)
(cid:0) (cid:1)
y2
1 216+126x+17x2 . (33)
− 12τ2(3+x)2
(cid:0) (cid:1)
The pressure p related to Φ [Eq. (23)] can be computed using either side of Eq. (22), viz.,
SHS SHS
∂Φ ∂Φ
SHS SHS
βp =n Φ + n = . (34)
SHS 0 SHS α
− ∂n ∂n
Xα α 3
With the advent of Eqs.(25) and (26) it canbe easily shown that p obeys the compressibilityequation, which for
SHS
the GPY closure is
∂βp
=a2, (35)
(cid:18) ∂ρ (cid:19)
T
where a is given by the definition in Eq. (8) [for details see Ref. 36]. Therefore, the pressure underlying the FMT is
always identical to the expression obtained from the GPY closure via the compressibility route. The quality of the
mMSA, C1, and PY pressures is discussed in Ref. 36 where the results are compared with computer simulations.
V. COMPARISON WITH SIMULATION DATA
A. Results for the radial distribution function
As a first test for the consistency of the density functional Ω, which uses the FMT excess free energy β [ ρ ]=
ex i
drΦ ( n (r) ) with Φ from Eq. (23) and the formulae φPY and φPY from Eqs. (27) and (28), wFe m{inim}ize
SHS { α } SHS 1 2
ΩR in a spherical geometry. A center particle identical to the fluid particles is realized via an external potential and
hence the resulting density profile correspondsto the RDF g(r) of the fluid up to a factor ρ. On the other hand, g(r)
can be obtained directly from the DCF c [Eq. (7) with Eq. (9)], using the OZ equation [Eq. (4)]. Obviously, for
PY
an exact DCF and an exact functional the two results must be equal, but given that approximations are involved on
both sides a deviation is to be expected. The degree of consistency and the agreement with numerical simulations
help evaluate the quality of the FMT constructed here.
For completeness, we include the formulae for the weighted densities in spherical geometry as they follow from
Eqs. (12) to (15). The result is
π r+R
n (r)= dr′r′ρ(r′) R2 (r r′)2 (36)
3
r Zr−R (cid:2) − − (cid:3)
2πR r+R
′ ′ ′
n (r)= dr r ρ(r ) (37)
2
r Zr−R
1 r+R R
′ ′ ′
n (r)= dr r ρ(r )+ [(r+R)ρ(r+R)+(r R)ρ(r R)] (38)
1
4r Zr−R 4r − −
1
n (r)= [(r+R/2)ρ(r+R)+(r R/2)ρ(r R)]
0
2r − −
R
′ ′
[(r+R)ρ(r+R) (r R)ρ(r R)] , (39)
− 4r − − −
where r, the distance from the center, is larger than R. Hence the problem of minimizing Ω in spherical geometry
can be reduced to a computation in 1D.
InFig.1wecomparetheFMTresultforg(r)fromthefullminimizationofΩtotheresultsobtained(a)byusingc
PY
intheOZequationand(b)fromMCsimulations. BoththesimulationdataandtheOZresultsaretakenfromRef.20.
7
a)
b)
1.25 MC 1.5 MC
η = 0.32 η = 0.4
FMT FMT
τ = 0.2 τ = 0.5
OZ OZ
1.25
g(r) g(r)
1
1
0.75 0.75
2 3 4 5 2 3 4 5
r/R r/R
c)
2.5
MC
η = 0.5
FMT
2 τ = 1
OZ
g(r)
1.5
1
2 3 4 5
r/R
FIG.1: RDFsg(r)ofSHSfluidswithdifferentBaxterparametersτ andpackingfractionsη. Thefulllinerepresentstheresult
from MC simulations [20] and the dotted line was calculated in Ref. 20 by using the PY result for the DCF, Eq. (7) with
Eq. (9), in the OZ integral equation. The dashed line is the result of the numerical minimization of the density functional Ω
which usesΦSHS from Eq.(23)and thecoefficients φP1Y andφP2Y from Eqs.(27) and(28). a) τ =0.2 and η=0.32. b)τ =0.5
and η=0.4. c) τ =1 and η=0.5.
For the three different choices of τ and η the FMT and OZ results are consistent with each other and both agree
well with the simulations. Deviations of the FMT results from the simulations occur mostly close to contact where
the theory either underestimates (Fig. 1a) or overestimates (Figs. 1b and c) the simulation data. Hence, there does
not appear to be a systematic deviation and the good agreement of the two theoretical results with the simulations
confirms the validity of the new functional.
B. Results for the inhomogeneous fluid
1. Density profiles
A standard approach to test a DFT’s ability to describe a fluid in an external potential which creates an inho-
mogeneity is to consider the fluid in contact with a planar hard wall. Due to the underlying symmetry the required
numerical minimization of the functional Ω is effectively a 1D computation and hence results are relatively easy to
obtain. The formulas for the weighted densities in the planar case can be obtained by considering Eqs. (36) to (39)
in the limit r . We compute density profiles for two different values of τ and three different bulk densities ρ
b
→ ∞
using the FMTs based on the PY, mMSA, and C1 closures for a fluid that is confined between two parallel walls at
a distance of 14R. In Fig. 2 the results are compared with data from canonical MC simulations taken from Ref. 40.
The PYbasedFMT providesan excellentagreementwith the simulationsand the C1 basedFMT performs nearlyas
8
0.1 0.2
a) ρR3 = 0.075 MC b) τ = 0.5 MC
b
0.075 0.15 FMT: PY C1
ρR3 = 0.05 τ = 0.2 mMSA
ρ(z)R3 0.05 b ρ(z)R3 0.1
ρR3 = 0.075
b
0.025 ρbR3 = 0.025 0.05 ρbR3 = 0.05
ρR3 = 0.025
b
FMT: PY C1 mMSA
0 0
0 1 2 3 4 5 6 0 1 2 3 4 5 6
z/R z/R
FIG.2: DensityprofilesofaSHSfluidconfinedbetween twoparallel hardwalls at adistance14R. Themidplaneis located at
z = 0. Simulation data (open circles) are taken from Ref. 40. The curves are the results from the DFT presented in Sec. IV.
The different bulk densities ρb of the fluid are given in the figure. a) Baxter’s parameter is τ =0.2. b) Baxter’s parameter is
τ =0.5.
well as the former. The mMSA based FMT shows deviations from the simulations which increase with decreasing τ
and increasing ρ . Taking into account that (a) the mMSA FMT approximates the cavity function at contact y (η)
b σ
by y (0) and (b) the C1 FMT incorporates the density dependence of y (η) only to lowest order in η, we conclude
σ σ
that the quality of the underlying approximation for y (η) is not the crucial factor that determines the quality of
σ
the FMT. Indeed, the key ingredient appears to be the structure of the DCF c (r) [Eq. (7)] that follows from the
GPY
GPYclosurerelation. ThequalityoftheFMTresultsisclearlysuperiortotheresultsobtainedfromtheOZequation
basedPYsingletapproximation[40,41]ortheDFTbyChoudhuryandGhosh[30],bothofwhichdeviatesignificantly
from simulations especially close to contact with the wall. Higher order DFTs [31–33] require a fitting parameter in
orderto matchthe correctcontactvalue orequationofstate. Incontrast,the factthatthe FMT respectsthe contact
theorem[42], i.e. βp=ρ , without an adjustable parameter,insures that the density profile is nicely describedby the
c
present theory even close to the wall.
2. Solvation force
As a final test of how well the FMT compares with simulations we consider the solvation force f between two
sol
parallel hard plates at a distance d. The solvation force (per unit area) is closely related to the contact value ρ of
c
the density profile via βf (d) = ρ (d) ρ ( ). Here, we choose to plot f (d) rather than ρ (d). This is because
sol c c sol c
− ∞
the accuracy of ρ ( ) is assured by the underlying equation of state through the contact theorem [42], while slight
c
∞
differencesintheequationofstateleadtodifferentoffsetsfortheρ (d)curveswhichobscuresthecomparisonbetween
c
the results. The solvation force for a fluid with Baxter parameter τ = 0.2 and bulk density ρ = 0.081/R3 is shown
b
in Fig. 3a. All the FMTs perform dramatically better than the OZ equation based singlet PY results [40] which are
particularly poor for the contact density. As expected, the mMSA FMT is inferior to the PY and C1 FMTs. In
contrast to the density profiles in Fig. 2, where the PY and C1 FMT both capture the simulation data equally well,
when describing f the PY FMT appears slightly superior to the C1 FMT. This is consistent with the fact that a
sol
better approximation to y underlies the PY FMT. Results for the more moderate case of a fluid which has τ = 0.5
σ
and ρ = 0.075/R3 are shown in Fig. 3b. The PY and C1 FMTs show excellent agreement with the simulation data
b
while the mMSA FMT are less accurate. The former versions of FMT clearly perform better than the DFT from
Ref. 33 (cf. figure 7 therein).
VI. CONCLUSION
We have constructeda DFT for the SHS fluid which, in the spirit of Rosenfeld’s FMT for the HS fluid [3], is based
onasetofweighteddensitieswhichcanbeidentifiedfromacarefulinspectionofthediluteHSfluid. Themainfinding
of this work is that using the FMT set of weighted densities (from the version of Kierlik and Rosinberg [37, 38]), in
9
0.01
a) ρR3 = 0.081 τ = 0.2 0.3 b) ρ R3 = 0.075 τ = 0.5
b b
0 MC
0.2
FMT PY
βfsolR3 MC βfsolR3 FMT C1
-0.01 0.1
FMT PY FMT mMSA
FMT C1
0
-0.02 FMT mMSA
0 1 2 3 4 5 6 0 1 2 3 4 5 6
d/R d/R
FIG. 3: Solvation force fsol between two hard plates as obtained in simulations [40] compared to the results from FMT. The
solvationforceiscalculatedfromthecontactvaluesofthedensityprofileaccordingtoβfsol(d)=ρc(d)−ρc(∞). a)TheBaxter
parameter is τ = 0.2 and the bulk density is ρb = 0.081/R3. b) The Baxter parameter is τ = 0.5 and the bulk density is
ρb=0.075/R3.
combination with an exact relation from SPT [34, 35], a density functional can be constructed which yields the PY
DCF [19] or the DCF resulting from any other closure of the broader GPY class [36]. The construction is uniquely
determined by the set of weighted densities [Eq. (11) with Eqs. (16) – (19)] the SPT differential equation [Eq. (22)]
and the given closure from the GPY class.
We tested the performance of the new FMT functional for the SHS fluid by comparison with simulation data for
the RDFand fordensity profilesofthe confinedfluidbetweentwo planarwalls. The agreementis verygoodwith the
mostsignificantimprovementoverprevioustheoriessuchasthe singletPYresults[40,41]or the DFT byChoudhury
and Ghosh [30] occurring in the density profiles close to contact with the wall. This reflects the property that the
FMT complies with the contact theorem [42], while other DFTs need an adjustable parameter to obtain the correct
contact value [32, 33]. The results for the solvation force between two planar plates are also significantly better than
the ones obtained with previous theories [33, 40].
Besides the obvious advantage of providing a very accurate description of the structure of the inhomogeneous SHS
fluid,theexistenceofanFMTformulationfortheSHSfluidhasseveralimportantimplicationsforfuturedevelopments.
One aspect is the extension of the theory to SHS mixtures. Given that FMT provides a simple and accurate theory
for HS mixtures, this new functional provides a starting point for the study of SHS mixtures. In particular, the
weighted densities are the key to the simplicity of the HS mixture FMT. However, unlike this situation, the new
functional depends on the particle radius R and the Baxter parameter τ. Hence, a straightforward generalization of
the functional with the help of weighted densities becomes impossible for mixtures in which R or τ or both R and τ
are different. A future challenge resides in seeking an appropriate set of weighted densities for the SHS fluid which
shifts the R and τ dependence from the functional to the weighted densities, thereby opening the door for broad
applicability to SHS mixtures. The complexion of such a set might be explored through incremental extension to
simple binary mixtures some of which allow for an analyticalsolution within the PY closure [22, 23]. Ultimately, one
might be able to treat non-sphericalsticky particles as was the case for hard particles [9].
AnadditionalfuturedevelopmentincludesextendingthepresentDFT formulationtofluidswithanattractionthat
has non-vanishing range such as the square-well fluid. Given that the PY closure cannot be solved analytically for
these systems, it seems most likely that a successful road to an FMT would include the creation of new weighted
densities whichaveragethe density profiles overa regionthat correspondsto the range of the attractivepotential. In
this context, studies of SW fluids with a very short but finite range of attraction [43] might provide a good starting
point.
Finally, there is the question whether the present FMT is suited for the study of SHS crystals. Problems such
as spurious divergences for peaked density distributions for HS systems lead us to believe that the SHS FMT in its
presentformcannotproperlydescribeaSHScrystal. However,theseshortcomingshavebeensuccessfullycuredusing
different approaches [6–8] within Rosenfeld’s formulation of FMT [3]. Nonetheless, these approaches have not yet
been translated to the KR framework [37] which constitutes the basis of the SHS FMT. The alternative approach of
formulating the present SHS FMT using Rosenfeld’s set of weighteddensities appears to be challenging. This follows
from the observation that the δ function in the DCF [Eq. (7)] can only be generated using KR’s weighted density
10
n . If the vectorial ~n in Rosenfeld’s FMT, supplemented by n and n , was used to construct Φ Φ the
1 2 3 2 SHS HS
−
DCF in Fourier space would acquire a term proportional to ~ω (k)~ω (k) [see Ref. 44 for the definition of ω~ ]. Such a
2 2 2
formulation is unable to yield the term proportional to sinkRcoskR/k originating from the δ-function in the DCF.
This term can only be matched if the product ω (k)ω (k), with KR’s ω (k) from Eq. (14), is used. Therefore, the
1 2 1
extensionof FMT to the SHS fluid providesanexample where the equivalence of the KR andRosenfeld formulations
isnolongervalid. Thisisaninterestingandunexpectedfinding thatillustratestheadvantageofoneformulationover
the other depending on the task at hand.
Acknowledgments
HHGisgratefultoRolandRothforstimulatingdiscussionsandtoMarkA.Millerforprovidingthesimulationdata
and Percus-Yevick results for the radial distribution functions. The authors acknowledge the U.S. National Science
Foundation Grant No. OPP0440841, the Department of Energy Grant No. DE-FG02-05ER15741, the Helmholtz
Gemeinschaft Alliance“Planetary Evolution and Life,”and Yale University for generous support of this research.
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